Methods and apparatus for conditioning communications signals based on detection of high-frequency events in polar domain

ABSTRACT

Methods and apparatus for conditioning communications signals based on detection of high-frequency in the polar domain. High-frequency events detected in a phase-difference component of a complex baseband signal in the polar domain are detected and used as a basis for performing hole-blowing on the complex baseband signal in the quadrature domain and/or nonlinear filtering either or both the magnitude and phase-difference components in the polar domain. Alternatively, high-frequency events detected in the phase-difference signal that correlate in time with low-magnitude events detected in a magnitude component of the complex baseband signal are used as a basis for performing hole-blowing on the complex baseband signal in the quadrature domain and/or nonlinear filtering either or both the magnitude and phase-difference components in the polar domain.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of U.S. patent applicationSer. No. 11/442,488, filed on May 26, 2006, now U.S. Pat. No. 7,675,993which is a continuation of U.S. patent application Ser. No. 10/037,870,filed on Oct. 22, 2001, now U.S. Pat. No. 7,054,385, both of which arehereby incorporated by reference.

FIELD OF THE INVENTION

The present invention relates to conditioning communications signals.More specifically, the present invention relates to conditioningcommunications signals based on detection of high-frequency events inthe polar domain.

STATE OF THE ART

Many modern digital radio communication systems transmit information byvarying both the magnitude and phase of an electromagnetic wave. Theprocess of translating information into the magnitude and phase of thetransmitted signal is typically referred to as modulation. Manydifferent modulation techniques are used in communication systems. Thechoice of modulation technique is typically influenced by thecomputational complexity needed to generate the signal, thecharacteristics of the radio channel, and, in mobile radio applications,the need for spectral efficiency, power efficiency and a small formfactor. Once a modulation technique has been selected for some specificapplication, it is oftentimes difficult or essentially impossible tochange the modulation. For example, in a cellular radio application allusers would be required to exchange their current mobile phone for a newphone designed to work with the new modulation technique. Clearly thisis not practical.

Many existing modulation formats have been designed to be transmittedwith radios that process the signal in rectangular coordinates. The twocomponents in the rectangular coordinate system are often referred to asthe in-phase and quadrature (I and Q) components. Such a transmitter isoften referred to as a quadrature modulator. (A distinction is drawnbetween a modulation, which is a mathematical description of the methodused to translate the information into the transmitted radio signal(e.g., BPSK, FSK, GMSK), and a modulator, which is the physical deviceused to perform this operation.) As an alternative, the transmitter mayprocess the signal in polar coordinates, in which case the signal isrepresented in terms of its magnitude and phase. In this case thetransmitter is said to employ a polar modulator. A polar modulator canhave several performance advantages over the more conventionalquadrature modulator, including higher signal fidelity, better spectralpurity, and lower dependence of device performance on temperaturevariation.

Although a polar modulator can have practical advantages over aquadrature modulator, the magnitude and phase components of the signaltypically have much higher bandwidth than the in-phase and quadraturecomponents. This bandwidth expansion has implications for digitalprocessing of the magnitude and phase, since the rate at which themagnitude and phase must be processed is dependent on their bandwidth.

The rate at which the magnitude and phase varies is very much dependenton the modulation technique. In particular, modulation formats that leadto very small magnitude values (relative to the average magnitude value)generally have very large phase component bandwidth. In fact, if thesignal magnitude goes to zero, the signal phase can instantaneouslychange by up to 180 degrees. In this case the bandwidth of the phasecomponent is essentially infinite, and the signal is not amenable totransmission by a polar modulator.

Many commonly employed modulation techniques do in fact lead to verysmall relative signal magnitude. To be more precise, theaverage-to-minimum signal magnitude ratio (AMR) is large. An importantpractical example of a modulation technique with large AMR is thetechnique employed in the UMTS 3GPP uplink (mobile-to-basestation).

Prior work in the field may be classified into two categories: one thatdeals generally with the reduction of peak power, and another that dealspecifically with “hole-blowing.” Hole-blowing refers to the process ofremoving low-power events in a communication signal that has atime-varying envelope. This name arises in that, using this technique, a“hole” appears in the vector diagram of a modified signal.

Much work has been done dealing with peak power reduction, in which thegoal is to locally reduce signal power. By contrast, relatively littlework appears to have been done that deals with hole-blowing (which seeksto locally increase signal power), and prior approaches have been foundto result in less-than-desired performance.

U.S. Pat. No. 5,805,640 (the '640 patent) entitled “Method and apparatusfor conditioning modulated signal for digital communications,” togetherwith U.S. Pat. No. 5,696,794 (the '794 patent), entitled “Method andapparatus for conditioning digitally modulated signals using channelsymbol adjustment,” both describe approaches for removing low magnitude(low power) events in communication signals. Both patents in fact referto creating “holes” in the signal constellation. The motivation givenfor creating these holes is that certain power amplifiers, in particularLINC power amplifiers, are difficult to implement when the signalamplitude dynamic range is large.

Briefly, the '794 patent teaches modifying the magnitude and phase ofthe symbols to be transmitted in order to maintain some minimum power.Since the symbols are modified before pulse shaping, the modified signalhas the same spectral properties as the original signal. The approachused in the '640 patent is to add a pulse having a certain magnitude andphase in between the original digital symbols before pulse shaping.Hence, whereas in the former patent data is processed at the symbol rate(T=1), in the latter patent, data is processed at twice the symbol rate(T=2). For brevity, these two methods will be referred to as the symbolrate method and the T/2 method, respectively. The method used tocalculate the magnitude and phase of the corrective pulse(s) is nearlyidentical in both patents.

Because the method used by both of these patents to calculate thecorrective magnitude and phase is only a very rough approximation,performance is less than desired. More particularly, after processingthe signal using either of these two approaches, the probability of alow power event is reduced, but remains significantly higher thandesired.

The specific approach used in the T/2 method is to add a pulse having aprescribed magnitude and phase to the signal at half-symbol timing(i.e., at t=k*T+T/2) before pulse shaping. The magnitude and phase ofthe additive pulse is designed to keep the signal magnitude fromdropping below some desired threshold. The method does not allow forplacement of pulses at arbitrary timing. As a result, effectiveness isdecreased, and EVM (error vector magnitude) suffers.

The method used in the T/2 approach to calculate the magnitude and phaseand of the additive pulse is very restrictive in that:

1) The signal envelope is only tested for a minimum value at half-symboltiming (t=i*T+T/2).

2) The phase of the correction is not based on the signal envelope, butrather only on the two symbols adjoining the low-magnitude event.

These two restrictions can lead to errors in the magnitude and phase ofthe corrective pulses. Specifically, the true signal minimum may occurnot at T/2, but at some slightly different time, so that error will beintroduced into the magnitude of the corrective pulses. The validity ofthis assumption is very much dependent on the specific signal modulationand pulse-shape. For example, this may be a reasonable assumption for aUMTS uplink signal with one DPDCH, but is not a reasonable assumptionfor a UMTS uplink signal with two DPDCH active. The size of thismagnitude error can be quite large. For example, in some cases themagnitude at T/2 is very near the desired minimum magnitude, but thetrue minimum is very close to zero. In such cases the calculatedcorrection magnitude is much smaller than would be desired, which inturn results in the low-magnitude event not being removed.

The signal envelope at T/2 may be greater than the desired minimum, butthe signal magnitude may be below the threshold during this inter-symboltime interval, so that a low-magnitude event may be missed entirely.

In any event, the correction magnitude obtained is often far from whatis needed.

The method used to calculate the corrective phase essentially assumesthat the phase of the pulse shaped waveform at T/2 will be very close tothe phase of straight line drawn between the adjoining symbols. This isan approximation in any case (although generally a reasonable one),which will introduce some error in the phase. However, thisapproximation is only valid if the origin does not lie between thepreviously described straight line and the true signal envelope. Whenthis assumption is violated, the corrective phase will be shifted byapproximately 180 degrees from the appropriate value. This typicallyleaves a low-magnitude event that is not corrected.

While the T/2 method adds pulses at half-symbol timing, the symbol ratemethod adds pulses to the two symbols that adjoin a low-magnitude event.That is, if the signal has a low-magnitude event at t=kT+T/2, thensymbols k and (k+1) will be modified. Both methods calculate the phaseof the additive pulses the same way, and both methods test for alow-magnitude event in the same way, i.e., the signal envelope athalf-symbol timing is tested. Therefore the same sources of magnitudeerror and phase error previously noted apply equally to this method.

The T/2 method applies the correction process repeatedly in an iterativefashion. That is, this method is applied iteratively “until there are nosymbol interval minima less than the minimum threshold”.

As compared to the foregoing methods, U.S. Pat. No. 5,727,026, “Methodand apparatus for peak suppression using complex scaling values,”addresses a distinctly different problem, namely reducing thePeak-to-Average power Ratio (PAR) of a communication signal. Large PARis a problem for many, if not most, conventional power amplifiers (PA).A signal with a large PAR requires highly linear amplification, which inturn affects the power efficiency of the PA. Reduction is accomplishedby adding a pulse to the original pulse-shaped waveform, with the pulsehaving an appropriate magnitude and phase such that the peak power isreduced. The pulse can be designed to have any desired spectralcharacteristics, so that the distortion can be kept in-band (to optimizeACPR), or allowed to leak somewhat out-of-band (to optimize EVM). Thetiming of the added pulse is dependent on the timing of the peak power,and is not constrained to lie at certain timing instants.

More particularly, this peak-reduction method adds a low-bandwidth pulseto the original (high PAR) signal. The added pulse is 180 degrees out ofphase with the signal at the peak magnitude, and the magnitude of theadditive pulse is the difference between the desired peak value and theactual peak value. Because the pulse is added to the signal (a linearoperation) the spectral properties of the additive pulse completelydetermine the effect of the peak reduction technique on the signalspectrum. The possibility is addressed of a peak value occurring at sometime that does not correspond to a sampling instant. The methodemphasizes the ability to control the amount of signal splatter and/orsignal distortion.

U.S. Pat. No. 6,175,551 also describes a method of PAR reduction,particularly for OFDM and multi-code CDMA signals where “a time-shiftedand scaled reference function is subtracted from a sampled signalinterval or symbol, such that each subtracted reference function reducesthe peak power”. The reference function is a windowed sinc function inthe preferred embodiment, or some other function that has “approximatelythe same bandwidth as the transmitted signal”.

Other patents of interest include U.S. Pat. Nos. 5,287,387; 5,727,026;5,930,299; 5,621,762; 5,381,449; 6,104,761; 6,147,984; 6,128,351;6,128,350; 6,125,103; 6,097,252; 5,838,724; 5,835,536; 5,835,816;5,838,724; 5,493,587; 5,384,547; 5,349,300; 5,302,914; 5,300,894; and4,410,955.

What is needed, and is not believed to be found in the prior art, is aprocess whereby the AMR of a communication signal can be greatly reducedwithout causing significant degradation to the signal quality.Desirably, this process would allow the practical implementation of apolar modulator even for signals that have very high AMR.

SUMMARY OF THE INVENTION

Methods and apparatus for conditioning communications signals based ondetection of high-frequency in the polar domain are disclosed. Anexemplary method includes setting a high-frequency threshold, convertingin-phase and quadrature components of a baseband signal into magnitudeand phase-difference components, and detecting high-frequency events inthe phase-difference component during which a rate of change of phase inthe phase-different component exceeds the high-frequency threshold. Thefrequency of the high-frequency threshold is set based on an in-bandperformance criterion (e.g., error vector magnitude (EVM)), anout-of-band performance criterion (e.g., adjacent channel leakage ratio(ACLR)), or on a balance or combination of in-band and out-of-bandperformance criteria. Once a high-frequency event is detected ahole-blowing process is performed on the in-phase and quadraturecomponents of the baseband signal in the quadrature domain.Additionally, or alternatively, nonlinear filtering is performed oneither or both the magnitude component and the phase-differencecomponent in the polar domain.

According to another exemplary method includes setting both ahigh-frequency threshold and a low-magnitude threshold, convertingin-phase and quadrature components of a baseband signal to magnitude andphase-difference components, and altering a signal trajectory of thebaseband signal (e.g., by performing a hole blowing process in thequadrature domain or nonlinear filtering the magnitude and/orphase-difference components in the polar domain), in response tolow-magnitude events in the magnitude component determined to fall belowthe low-magnitude threshold and high-frequency events in thephase-difference component determined to both exceed the high-frequencythreshold and correlate in time with the detected low-magnitude eventsin the magnitude component. The frequency of the high-frequencythreshold and magnitude of the low-magnitude threshold are set based onan in-band performance criterion (e.g., EVM), an out-of-band performancecriterion (e.g., ACLR), or on a balance or combination of in-band andout-of-band performance criteria.

Further features and advantages of the present invention, including adescription of the structure and operation of the above-summarized andother exemplary embodiments of the invention, are described in detailbelow with respect to accompanying drawings, in which like referencenumbers are used to indicate identical or functionally similar elements.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention may be further understood from the followingdescription in conjunction with the appended drawing. In the drawings:

FIG. 1 is a generic block diagram for generation of a PAM signal;

FIG. 2 a is a block diagram of a PAM generator modified for AMRreduction;

FIG. 2 b is a more detailed block diagram of an apparatus like that ofFIG. 2 a;

FIG. 3 shows the impulse response of a square-root raised-cosine pulseshaping filter with 22% excess bandwidth;

FIG. 4 shows an I-Q plot (vector diagram) of a portion of a QPSK signalwith square-root raised-cosine pulse shaping;

FIG. 5 shows a section of the power of the pulse-shaped QPSK signal as afunction of time;

FIG. 6 is a histogram showing the temporal location of signal magnitudeminima, including only those minima that are more than 12 dB below themean power;

FIG. 7 shows phase of the QPSK signal as a function of time;

FIG. 8 shows instantaneous frequency expressed as a multiple of thesymbol rate for the signal of FIG. 7;

FIG. 9 a illustrates a situation in which a mathematical model may beused to detect the occurrence of a low magnitude event;

FIG. 9 b is a geometric illustration of the locally linear model for thecomplex signal envelope;

FIG. 9 c illustrates calculation of t_min based on the locally linearmodel;

FIG. 10 shows a comparison of the original signal power and the power ofthe signal after addition of a single, complex-weighted pulse designedto keep the minimum power above-12 dB of average power;

FIG. 11 is an I-Q plot of a QPSK signal that has been modified to keepthe instantaneous power greater than −12 dB relative to the RMS power;

FIG. 12 shows the instantaneous frequency of the modified signal,expressed as a multiple of the symbol rate;

FIG. 13 shows minimum power in the modified signal as a function oferror in the time at which the corrective pulse is added (the magnitudeand phase of the corrective pulse are held constant);

FIG. 14 shows estimated PSD of a conventional QPSK signal and the samesignal after application of the exact hole-blowing method;

FIG. 15 a is an I-Q plot of the modified signal after demodulation,showing that some distortion has been introduced into the signal (themeasured RMS EVM is 6.3%);

FIG. 15 b illustrates the results of non-linear filtering using aroot-raised cosine pulse that is the same as the pulse-shaping filter;

FIG. 15 c illustrates the results of non-linear filtering using aHanning window for the correction pulse, with the time duration equal to½ the symbol duration;

FIG. 16 a is a block diagram illustrating symbol-rate hole-blowing;

FIG. 16 b is a more detailed block diagram illustrating symbol-ratehole-blowing;

FIG. 16 c is a block diagram illustrating the iteration of symbol-ratehole-blowing;

FIG. 16 d is a block diagram illustrating the iteration of symbol-ratehole-blowing;

FIG. 16 e is a block diagram illustrating the concatenation of one ormore iterations of symbol-rate hole-blowing followed by one or moreiterations of symbol-rate hole-blowing;

FIG. 17 a is a block diagram of a portion of a radio transmitter inwhich polar domain nonlinear filtering is performed;

FIG. 17 b is a more detailed block diagram of an apparatus like that ofFIG. 17;

FIG. 18 is a waveform diagram showing the magnitude component of thepolar coordinate signal and showing the difference of the phasecomponent of the polar coordinate signal;

FIG. 19 is a diagram of the impulse response of a DZ3 pulse;

FIG. 20 is a waveform diagram showing results of nonlinear filtering ofthe magnitude component (showing the original magnitude component beforepolar domain nonlinear filtering, the magnitude component after thepolar domain nonlinear filtering, the threshold, and the pulses added tothe magnitude component);

FIG. 21 shows an example of an added pulse suitable for nonlinearfiltering of the phase component;

FIG. 22 is a waveform diagram showing results of nonlinear filtering ofthe phase-difference component;

FIG. 23 illustrates an alternate way for nonlinear filtering of thephase component in a polar coordinate system;

FIG. 24 is a block diagram of a portion of a radio transmitter in whichnonlinear filtering is performed first in the quadrature domain and thenin the polar domain;

FIG. 25 is a PSD showing results of the nonlinear filtering of FIG. 34.

FIG. 26 is an I-Q plot for the UMTS signal constellation with one activedata channel and a beta ratio of 7/15;

FIG. 27 is an I-Q plot for the UMTS signal constellation with two activedata channels and a beta ratio of 7/15;

FIG. 28 a illustrates the manner of finding the timings of low-magnitudeevents;

FIG. 28 b compares probability density functions of low magnitude eventsusing the exact algorithm and using the real-time approximation;

FIG. 29 is an I-Q plot illustrating a line-comparison method for vectorquantization;

FIG. 30 illustrates a CORDIC-like algorithm for vector quantization.

FIG. 31 is an illustration of a known method of calculating the phase ofthe corrective pulse(s);

FIG. 32 shows Cumulative Distribution Functions (CDF) obtained with thepresent method and with known methods when the signal is pi/4 QPSK withraised cosine pulse-shaping and excess bandwidth of 22% (the desiredminimum power is 9 dB below RMS);

FIG. 33 illustrates an example where the known symbol rate method worksfairly well (the original signal envelope is shown, the modifiedenvelope is shown, and the sample used to calculate the correctionmagnitude is indicated);

FIG. 34 illustrates an example where the known symbol rate method doesnot work well;

FIG. 35 illustrates an example where the known T/2 method does not workwell;

FIG. 36 shows Cumulative Distribution Functions (CDF) obtained with thepresent method and the two known methods when the signal is a 3GPPuplink signal with one active DPDCH and amplitude ratio of 7/15 (thedesired minimum power is 9 dB below RMS);

FIG. 37 is a waveform diagram of the magnitude component of apolar-coordinate baseband signal, highlighting the fact that somelow-magnitude events do not correspond to high-frequency events;

FIG. 38 is a waveform diagram of the phase-difference component of apolar-coordinate baseband signal, highlighting occurrences ofhigh-frequency events;

FIG. 39 is a block diagram of an apparatus for detecting high-frequencyevents in the phase-difference component shown in FIG. 38, according toan embodiment of the present invention;

FIG. 40 is a waveform diagram of the magnitude and phase-differencecomponents of a polar-coordinate baseband signal; and

FIG. 41 is a block diagram of an apparatus for detecting low-magnitudeevents in the magnitude component shown in FIG. 40 and time-correlatedhigh-frequency events in the phase-difference component shown in FIG.40.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

A polar modulator can be viewed as a combination of a phase modulatorand an amplitude modulator. The demands placed on the phase modulatorand amplitude modulator are directly dependent on the bandwidth of thesignal's phase and magnitude components, respectively. The magnitude andphase bandwidth, in turn, are dependent on the average-to-minimummagnitude ratio (AMR) of the signal. As will be shown later, a signalwith large AMR can have very abrupt changes in phase, which means thatthe signal phase component has significant high frequency content.Furthermore, certain transistor technologies limit the AMR that can beachieved in a practical amplitude modulator. This limitation can lead todistortion of the transmitted signal if the required magnitude dynamicrange exceeds that which can be generated by the transistor circuit.Thus minimization of signal AMR is highly desirable if the signal is tobe transmitted with a polar modulator. One example of a polar modulatoris described in U.S. Pat. No. 7,158,494 entitled “Multi-modecommunications transmitter,” filed on even date herewith andincorporated herein by reference.

The non-linear digital signal processing techniques described hereinmodify the magnitude and phase of a communication signal in order toease the implementation of a polar modulator. Specifically, themagnitude of the modified signal is constrained to fall within a certaindesired range of values. This constraint results in lower AMR comparedto the original signal, which in turn reduces the magnitude and phasebandwidth. The cost of this reduction in bandwidth is lower signalquality. However, the reduction in signal quality is generally small,such that the final signal quality is more than adequate.

Signal quality requirements can typically be divided into in-band andout-of-band requirements. Specifications that deal with in-band signalquality generally ensure that an intended receiver will be able toextract the message sent by the transmitter, whether that message bevoice, video, or data. Specifications that deal with out-of-band signalquality generally ensure that the transmitter does not interfereexcessively with receivers other than the intended receivers.

The conventional in-band quality measure is the RMS error vectormagnitude (EVM). A mathematically related measure is rho, which is thenormalized cross-correlation coefficient between the transmitted signaland its ideal version. The EVM and rho relate to the ease with which anintended receiver can extract the message from the transmitted signal.As EVM increases above zero, or rho decreases below one, the transmittedsignal is increasingly distorted relative to the ideal signal. Thisdistortion increases the likelihood that the receiver will make errorswhile extracting the message.

The conventional out-of-band quality measure is the power spectraldensity (PSD) of the transmitted signal, or some measure derivedtherefrom such as ACLR, ACP, etc. Of particular interest in relation toPSD is the degree to which the transmitted signal interferes with otherradio channels. In a wireless communications network, interference withother radio channels reduces the overall capacity of the network (e.g.,the number of simultaneous users is reduced).

It should be clear that any means of reducing the average-to-minimummagnitude ratio (AMR) must create as little interference as possible(minimal degradation to out-of-band signal quality) while simultaneouslymaintaining the in-band measure of signal quality (i.e., EVM or rho) atan acceptable level. These considerations motivate the presentinvention, which reduces AMR while preserving out-of-band signalquality, which is of particular importance to operators of wirelesscommunications networks.

In general, AMR reduction is performed by analyzing the signal to betransmitted, and adding carefully formed pulses into the signal in timeintervals in which the signal magnitude is smaller than some threshold.The details of an exemplary embodiment, including the signal analysisand pulse formation, are described below, starting with a description ofa class of signals for which the invention is useful.

Pulse-Amplitude Modulation (PAM)

Many modern communication systems transmit digital messages using ascheme called pulse amplitude modulation (PAM). A PAM signal is merely afrequency-upconverted sum of amplitude-scaled, phase-shifted, andtime-shifted versions of a single pulse. The amplitude-scaling andphase-shifting of the n^(th) time-shifted version of the pulse aredetermined by the n^(th) component of the digital message. In the fieldof communication systems, the broad class of PAM signals includessignals commonly referred to as PAM, QAM, and PSK, and many variantsthereof. Mathematically, a PAM signal x(t) at time t can be described asfollows, as will be recognized by those skilled in the art ofcommunications theory. The description is given in two parts, namely thefrequency-upconversion and amplification process, and the basebandmodulation process, as shown in FIG. 1.

The frequency-upconversion and amplification process can be describedmathematically as follows:x(t)=Re{gs(t)e ^(jw) ^(c) ^(t))}where Re{ } denotes the real part of its complex argument, ω_(c)=2πf_(c)defines the radio carrier frequency in radians per second and Hz,respectively, j is the imaginary square-root of negative unity, and g isthe amplifier gain. This equation describes the frequency-upconversionprocess used to frequency-upconvert and amplify the complex basebandsignal s(t), which is also the so-called I/Q (in-phase/quadrature)representation of the signal. The signal s(t), which is created by thebaseband modulation process, is defined mathematically by

${s(t)} = {\sum\limits_{n}{a_{n}{p\left( {t - {nT}} \right)}}}$where p(t) is the pulse at time t and T is the symbol period (1/T is thesymbol rate). For any time instant t at which s(t) is desired, thesummation is taken over all values of n at which p(t−nT) isnon-negligible. Also, a_(n) is the symbol corresponding to the n^(th)component of the digital message. The symbol a_(n) can be either real orcomplex, and can be obtained from the n^(th) component of the digitalmessage by means of either a fixed mapping or a time-variant mapping. Anexample of a fixed mapping occurs for QPSK signals, in which the n^(th)component of the digital message is an integer d_(n) in the set {0, 1,2, 3} and the mapping is given by a_(n)=exp(jπd_(n)/2). An example of atime-variant mapping occurs for π/4-shifted QPSK, which uses a modifiedQPSK mapping given by a_(n)=exp(jπ(n+2d _(n))/4); that is, the mappingdepends on the time-index n, not only on the message value d_(n).

For the present invention, an important property of PAM signals is thatthe shape of the PSD (as a function of f) of a PAM signal is determinedexclusively by the pulse p(t), under the assumption that the symbolsequence an has the same second-order statistical properties as whitenoise. This property may be appreciated by considering the signal s(t)as the output of a filter having impulse response p(t) and being drivenby a sequence of impulses with weights a_(n). That is, the PSD S_(x)(f)of x(t) can be shown to be equal to

${S_{x}(f)} = {\frac{g^{2}\sigma_{a}^{2}}{4\; T}\left( {{{P\left( {f - f_{c}} \right)}}^{2} + {{P\left( {f + f_{c}} \right)}}^{2}} \right)}$where P(f) is the Fourier transform of the pulse p(t), and σ_(a) ² isthe mean-square value of the symbol sequence.

This important observation motivates the present method, because itsuggests that adding extra copies of the pulse into s(t) does not alterthe shape of the PSD. That is, nonlinear filtering performed in thismanner can result in not just minor but in fact imperceptible changes inPSD. The adding of extra copies of the pulse into the signal can be usedto increase the amplitude of x(t) as desired, for example when it fallsbelow some threshold. Specifically, s(t) may be modified by addingadditional pulses to it, to form new signals ŝ(t) and {circumflex over(x)}(t):

x̂(t) = Re{g ŝ(t)𝕖^(j w_(c)t)} where${{\hat{s}(t)} = {{\sum\limits_{n}{a_{n}{p\left( {t - {nT}} \right)}}} + {\sum\limits_{m}{b_{m}{p\left( {t - t_{m}} \right)}}}}},$and the perturbation instances t_(m) occur at points in time when it isdesired to perturb the signal (e.g., whenever the magnitude of s(t)falls below some threshold). The perturbation sequence b_(m) representsthe amplitude-scaling and phase-shifting to be applied to the pulsecentered at time t_(m) (e.g., chosen so as to increase the magnitude ofs(t) in the vicinity of time t_(m)). Like the first term in ŝ(t), thesecond term in ŝ(t) can be thought of as the output of a filter havingimpulse response p(t) and being driven by a sequence of impulses withweights b_(m). Thus, it is reasonable to expect that the PSDs of{circumflex over (x)}(t) and {circumflex over (x)}(t) will have verysimilar shapes (as a function of frequency f).

With this theoretical underpinning, the present invention can bedescribed in detail, in a slightly more general form than used above, asdepicted in FIG. 2 a. The invention takes the signal s(t) as its input.This signal passes into an analyzer, which determines appropriateperturbation instances t_(m), and outputs a perturbation sequence valueb_(m) at time instant t_(m). The perturbation sequence passes through apulse-shaping filter with impulse response r(t), the output of which isadded to s(t) to produce ŝ(t), which in turn is passed to anyappropriate means for frequency upconversion and amplification to thedesired power. The pulse-shaping filter r(t) can be identical to theoriginal pulse p(t), as described above, or it can be different fromp(t) (e.g., it may be a truncated version of p(t) to simplifyimplementation).

A more detailed block diagram is shown in FIG. 2 b, illustrating a mainsignal path and a correction signal path for two signal channels, I andQ. Pulse-shaping may occur after pulse-shaping (sample-rate correction)or prior to pulse-shaping (symbol-rate correction). In the correctionpath, sequential values of I and Q are used to perform a signal minimumcalculation and comparison with a desired minimum. If correction isrequired based on the comparison results, then for each channel themagnitude of the required correction is calculated. A pulse (which maybe the same as that used for pulse-shaping) is scaled according to therequired correction and added into the channel of the main path, whichwill have been delayed to allow time for the correction operations to beperformed.

The method used to determine the timing, magnitude, and phase of thecorrection pulses is dependent on the modulation format. Factors toconsider include:

1. The duration of low-magnitude events relative to the symbol period.

2. The timing distribution of low-magnitude events.

If the duration of all low-magnitude events are small relative to thesymbol (or chip) duration, then each low-magnitude event can becorrected by the addition of a single complex-weighted pulse, which canbe identical to that used in pulse-shaping. This approach would beeffective with, e.g., M-ary PSK modulation. The appropriate hole-blowingmethod in this case is referred to as the “exact” hole-blowing method.The exact hole-blowing method is described below, as well as a practicalreal-time hardware implementation of the same. Other modulation formatsmay lead to low-magnitude events of relatively long duration. This wouldtypically be the case with, e.g., QAM and multi-code CDMA modulation. Insuch cases multiple pulses may be added, or, alternatively, multipleiterations of the exact hole-blowing method be used. The usefulness ofpolar-domain hole-blowing has been demonstrated to perform “finalclean-up” of a signal produced using one of the foregoing techniques,achieving even better EVM performance. Because magnitude information isavailable explicitly in the polar-domain representation, hole-blowingmay be performed on a sample-by-sample basis, as compared to (typically)a symbol-by-symbol (or chip) basis in the case of the foregoingtechniques.The Exact Hole-Blowing Method

The detailed operation of the exact hole-blowing method is nowdemonstrated by example. A QPSK signal with square-root raised-cosinepulse shaping is used for this example. The pulse shaping filter has 22%excess bandwidth, and is shown in FIG. 3. A typical I/Q plot of thissignal is presented in FIG. 4. Clearly the signal magnitude can becomearbitrarily small. The signal power over a short period of time is shownin FIG. 5. The average power is normalized to unity (0 dB). This figureindicates that the AMR of this signal is at least 40 dB. In reality, theAMR for this QPSK signal is effectively infinite, since the signal powercan become arbitrarily small. The timing of the minimum power isimportant, since the time instants at which to insert the correctionpulses must be determined. One would expect that the power minima wouldapproximately occur at t=nT/2, where n is an integer and T is the symbolperiod. This is supported by FIG. 5, which shows that the minimum poweroccurs very near T/2.

To further support this hypothesis, the distribution of the power minimatiming was examined. For this purpose, a pulse-shaped QPSK waveform withrandom message was generated, and the timing at which the power minimaoccurred was determined. For this example, it is assumed that a lowpower event occurs if the instantaneous signal power was more than 12 dBbelow the mean signal power.

FIG. 6 shows a histogram of the timing of the QPSK signal magnitudeminima. These results are based on 16384 independent identicallydistributed symbols. Note that the minima are indeed closely clusteredaround a symbol timing of T/2, as expected. This is an important result,since it limits the range over which the search for a signal minimummust be performed. (It should be noted that the histogram shown in FIG.6 is only valid for this specific signal type (QPSK). Other signaltypes, such as high order QAM, may have different distributions, whichmust be taken into account in the search for local power-minima.)

As stated previously, low-power events are associated with rapid changesin the signal phase. This correspondence is illustrated in FIG. 7, whichshows the phase of the QPSK signal corresponding to the power profileshown in FIG. 5. It is clear that the phase changes rapidly near t=T/2,corresponding to the minimum power. This characteristic can be seen moreexplicitly in FIG. 8, which shows the instantaneous frequency, definedhere for the sampled data waveform as

${f_{i}(t)} = \frac{{\theta\left( {t + \delta} \right)} - {\theta(t)}}{2{\pi\delta}}$where θ(t) is the signal phase at time instant t and δ is the samplingperiod. FIG. 8 shows that the instantaneous frequency over this intervalis up to 45 times the symbol rate. To put this in perspective, the chiprate for the UMTS 3GPP wideband CDMA standard is 3.84 MHz. If the symbolrate for the QPSK signal in our example was 3.84 MHz, the instantaneousfrequency would exceed 45×3.84=172.8 MHz. Processing a signal with suchhigh instantaneous frequency is not yet practical.

It is apparent that the bandwidth of the signal phase must be reduced inorder to enable practical implementation of a polar modulator. The mostobvious approach is to simply low-pass filter the phase (or equivalentlythe phase difference). However, any substantial filtering of thephase-difference will lead to unacceptably large non-linear distortionof the signal. This distortion in turn leads to a large increase inout-of-band signal energy, which is typically not acceptable. Insteadthe recognition is made that rapid changes in signal phase only occurwhen the signal magnitude is very small. Therefore if the signalmagnitude can be kept above some minimum value, the bandwidth of thesignal phase will be reduced. As should be evident from the discussionof PAM signal spectral properties, the signal can be modified by theaddition of carefully selected pulses without any apparent effect on thesignal bandwidth.

In order to avoid low-magnitude events, and consequently reduce theinstantaneous frequency, a complex weighted version of the pulse-shapingfilter is added to the signal. This complex weighted pulse is referredto as the correction pulse. The phase of the pulse is selected so thatit combines coherently with the signal at the point of minimummagnitude. The magnitude of the correction pulse is equal to thedifference between the desired minimum magnitude and the actual minimummagnitude of the signal.

Care must be taken in the calculation of the correction magnitude andphase in order to obtain the desired effect. Most importantly, it mustbe recognized that the minimum signal magnitude may not correspond to asampling instant. Since the phase and magnitude are changing rapidly inthe neighborhood of a local power minimum, choosing the correction phasebased on a signal value that does not correspond to the signal minimumcan lead to large phase and/or magnitude error. The likelihood of alarge error increases as the minimum magnitude grows smaller. Thelikelihood of a large error can be lessened by using a very highsampling rate (i.e., a large number of samples per symbol), but thisgreatly, and unnecessarily, increases the computational load.

Instead, a so-called “locally linear model” is fitted to the signal inthe temporal neighborhood of a local power minimum, and the minimummagnitude of the model is solved for mathematically. The locally linearmodel effectively interpolates the signal so that the magnitude can becalculated directly at any arbitrary time instant. In this way thecalculation of the corrective pulses are not limited to the valuespresent in the sampled data waveform. This allows for highly accuratecalculation of the minimum magnitude and, of equal importance, therequired correction phase.

Note that the complete pulse-shaped signal need not exist in order tocalculate the correction magnitude and phase. The complex basebandsignal need only be calculated at a small number of key samplinginstants, allowing the non-linear filtering to operate on the rawsymbols before a final pulse-shaping step. This expedient can bebeneficial in real-time implementations.

A signal may lie outside the exclusion region defined by the desiredminimum at two points fairly near in time, yet pass through theexclusion region in between the two points, as illustrated in FIG. 9 a.In order to find the true minimum value of the signal magnitude, and itscorresponding phase, the following approach is preferably used. Thisapproach employs a locally linear model for the complex baseband signalenvelope, as illustrated in FIG. 9 b. The model approximates the complexsignal envelope by a straight line in I/Q space in the vicinity of thelocal power minimum. In many cases, if the desired minimum power issmall, and the signal modulation is not too complicated (e.g., PSK),this model is quite accurate. Once this model has been fitted to thesignal, the minimum magnitude of the locally linear model can be solvedfor directly.

In order to formulate the model, the complex signal must be known at noless than two distinct time instants. These time instants willpreferably be close to the true minimum value of the signal, since theaccuracy of the model is better in this case. These time instants neednot correspond to sampling instants that are present in the sampledsignal envelope.

Denote the complex signal at time t1 by s(t1). Denote the real andimaginary parts of s(t1) by x1 and y1, respectively. Similarly, denotethe real and imaginary parts of s(t2) by x2 and y2, respectively. Thesignal at these time instants are given by:

${s\left( {t\; 1} \right)} = {\sum\limits_{i}{a_{i}{p\left( {{t\; 1} - {{\mathbb{i}}\; T}} \right)}}}$${s\left( {t\; 2} \right)} = {\sum\limits_{i}{a_{i}{p\left( {{t\; 2} - {{\mathbb{i}}\; T}} \right)}}}$where t2>t1, and t2−t1>>T. Then defineΔx=x2−x1Δy=y2−y1

To the degree that the locally linear model is accurate, the complexsignal at any instant in time t can be represented ass(t)=s(t1)+cΔx+jcΔyRe{s(t)}x1=cΔxIm{s(t)}=y1+cΔywhere c is a slope parameter.

In order to find the minimum magnitude of this linear model, a geometricapproach based on FIG. 9 b is used. The point of minimum magnitudecorresponds to a point on the locally linear model that intersects asecond line which passes through the origin and is orthogonal to thelinear model, as illustrated in FIG. 9 b. This orthogonal line can bedescribed parametrically (using the parameter g) by the equations:x=−gΔyy=gΔx

The point of intersection is found by setting the x-axis and y-axiscomponents of the linear model and the orthogonal line equal. Thisyields the following set of equations (two equations and two unknowns):x1+cΔx=−gΔyy1+cΔy=gΔxRe-arranging these equations gives:

$c = \frac{- \left( {{g\;\Delta\; y} + {x\; 1}} \right)}{\Delta\; x}$$c = \left( \frac{{g\;\Delta\; x} - {y\; 1}}{\Delta\; y} \right)$Setting these two expressions equal, and solving for g, gives:

$g = \frac{\left( {{y\; 1\Delta\; x} - {x\; 1\Delta\; y}} \right)}{{\Delta\; x^{2}} + {\Delta\; y^{2}}}$The minimum magnitude of the linear model is then

$\quad{\quad\begin{matrix}{\rho_{\min}^{2} = {\left( {g\;\Delta\; x} \right)^{2} + \left( {g\;\Delta\; y} \right)^{2}}} \\{\rho_{\min} = {{g}\sqrt{{\Delta\; x^{2}} + {\Delta\; y^{2}}}}} \\{= \frac{{{y\; 1\;\Delta\; x} - {x\; 1\Delta\; y}}}{\sqrt{{\Delta\; x^{2}} + {\Delta\; y^{2}}}}}\end{matrix}}$

The minimum magnitude as calculated from the locally linear model mustbe explicitly calculated in order to test for a low-magnitude event andto calculate the correction magnitude. The minimum magnitude is alsoused to calculate the magnitude of the correction pulse, which is givenby:ρ_(cprr)=ρ_(desired)−ρ_(min)

The correction phase is equal to the phase of the signal at minimummagnitude, as determined from the locally linear model. If thecorrections are being performed in the I/Q domain, there is no need toexplicitly calculate the phase θ of the correction—only sin(θ) andcos(θ) need be calculated. Referring to FIG. 9 b, it may be seen that:

$\quad\begin{matrix}{{\rho_{\min}\cos\;\theta} = {{- g}\;\Delta\; y}} \\{{\cos\;\theta} = \frac{{- g}\;\Delta\; y}{\rho_{\min}}} \\{= \frac{{- \Delta}\;{y\left( {{y\; 1\;\Delta\; x} - {x\; 1\Delta\; y}} \right)}\sqrt{{\Delta\; x^{2}} + {\Delta\; y^{2}}}}{{{{y\; 1\Delta\; x} - {x\; 1\Delta\; y}}}\left( {{\Delta\; x^{2}} + {\Delta\; y^{2}}} \right)}} \\{= \frac{{- \Delta}\; y\;{{sign}\left( {{y\; 1\Delta\; x} - {x\; 1\Delta\; y}} \right)}}{\sqrt{{\Delta\; x^{2}} + {\Delta\; y^{2}}}}}\end{matrix}$where c/|c|=sign(c) for any scalar C. Similarly:

${\sin\;\theta} = \frac{\Delta\; x\;{{sign}\left( {{y\; 1\Delta\; x} - {x\; 1\Delta\; y}} \right)}}{\sqrt{{\Delta\; x^{2}} + {\Delta\; y^{2}}}}$In rectangular coordinates, the in-phase and quadrature correctionfactors will respectively have the form:c _(I)=(ρ_(desired)−ρ_(min))cos θc _(Q)=(ρ_(desired)−ρ_(min))sin θThus, it is seen that cos θ and sin θ are sufficient in order tocalculate the correction factors, and the independent determination ofsignal phase θ is not necessary. The modified signal in rectangularcoordinates is given by:{circumflex over (s)}_(I)(t)=s _(I)(t)+c _(I) p(t−t _(min)){circumflex over (s)}_(Q)(t)=s _(Q)(t)+c _(Q) p(t−t _(min))where t_min is the approximate time at which the local minimum occurs,and the pulse is assumed (without loss of generality) to be normalizedsuch that p(0)=1.

In some applications it may be desirable to obtain a precise estimate ofthe time t_min corresponding to the minimum signal power. Here,“precise” is used to mean an estimate that is not limited by the samplerate, and may have an arbitrary degree of accuracy. For example, givent_min, correction pulses may be added to the signal based on aninterpolated or highly oversampled prototype pulse p(t). For thispurpose, any of the three following procedures that exploit the locallylinear model can be used. The first two approaches are based ongeometric arguments. The third approach is based on direct minimizationof the signal magnitude.

Linear interpolation is typically employed to estimate a signal value atsome arbitrary time t when the signal is only known at two or morediscrete time indices. That is, linear interpolation is used to finds(t) given t. However, linear interpolation can also be used to find tgiven s(t). This property may be exploited as follows, in accordancewith the illustration of FIG. 9 c. The linear model is

s(t) = s(t 1) + c(Δ x + j Δ y) where$c = \frac{t - {t\; 1}}{{t\; 2} - {t\; 1}}$For the real part of the signal,

$x = {x_{1} + {\frac{t - t_{1}}{t_{2} - t_{1}}\left( {x_{2} - x_{1}} \right)}}$Solving this expression for t,

$t = {t_{1} + {\frac{x - x_{1}}{x_{2} - x_{1}}\left( {t_{2} - t_{1}} \right)}}$

The calculation of t_min depends on the calculation of x_min, the realpart of the signal at the local minimum magnitude. As describedpreviously, the minimum magnitude of the signal is ρ_(min); the realpart of the signal at minimum magnitude is ρ_(min) cos θ. Therefore

$t_{\min} = {t_{1} + {\left( \frac{{\rho_{\min}\cos\;\theta} - x_{1}}{x_{2} - x_{1}} \right)\left( {t_{2} - t_{1}} \right)}}$Based on similar arguments for the imaginary part of the signal,alternatively

$t_{m\; i\; n} = {t_{1} + {\left( \frac{{\rho_{m\; i\; n}\sin\;\theta} - y_{1}}{y_{2} - y_{1}} \right)\left( {t_{2} - t_{1}} \right)}}$

The two expressions described above only depend on the real part orimaginary part of the signal. In a finite precision implementation, thismay be a drawback if the change in x or y is small. Therefore it may bebetter in practice to use an expression which is dependent on both thereal and imaginary parts of the signal. Such an expression can bederived by direct minimization of the signal magnitude using standardoptimization procedures. Using the linear model, the signal magnitude is|s(t)|²=(x ₁ +cΔx)²+(y ₁ +cΔy)²Taking the derivative with respect to c,

${\frac{\partial}{\partial c}{{s(t)}}^{2}} = {{2\left( {x_{1} + {c\;\Delta_{x}}} \right)\Delta_{x}} + {2\left( {{y\; 1} + {c\;{\Delta\;}_{y}}} \right)\Delta_{y}}}$Setting the derivative equal to zero and solving for c,

$c_{m\; i\; n} = \frac{- \left( {{x_{1}\Delta_{x}} + {y_{1}\Delta_{y}}} \right)}{\Delta_{x}^{2} + \Delta_{y}^{2}}$Thus the final expression for t_min is

$t_{m\; i\; n} = {t_{1} + {\left( \frac{- \left( {{x_{1}\Delta_{x}} + {y_{1}{\Delta\;}_{y}}} \right)}{\Delta_{x}^{2} + \Delta_{y}^{2}} \right)\left( {t_{2} - t_{1}} \right)}}$The “exact” hole-blowing algorithm can thus be summarized as follows:1. Determine the approximate timing t=t1 of a potential low-magnitudeevent in the signal.2. In the temporal neighborhood of a potential low-magnitude event,calculate the pulse-shaped signal s(t) for at least two distinct timeinstants t1 and t2>t1, where t2-t2>>T. In the case of symbol-ratehole-blowing, the signal s(t) is calculated based on thelater-to-be-applied bandlimiting pulse and some number of symbols in thevicinity of t1 and some number of symbols in the vicinity of t2. In thecase of sample-rate (i.e., oversampled) hole-blowing, pulseshaping willhave already been performed, such that s(t1) and s(t2) may be chosen tocorrespond with adjacent samples.3. Calculate the minimum magnitude ρ_(min) using the “locally-linearmodel”, detailed above, and t_(min), the time of ρ_(min).4. Compare the calculated minimum magnitude to the desired minimummagnitude.5. If the calculated minimum magnitude is less than the desired minimummagnitude, calculate the in-phase and quadrature correction weightsc_(I) and c_(Q), respectively, as detailed above.6. Weight two copies of the pulse-shaping filter by the in-phase andquadrature correction values, respectively.7. Add these weighted copies of the pulse-shaping filter to the in-phaseand quadrature components of the signal, referenced to t_(min).8. Translate the modified in-phase and quadrature components tomagnitude and phase, forming the signals to be processed by a polarmodulator.

The effectiveness of the exact hole-blowing method described above isevident in FIG. 10, comparing the instantaneous power of the originalQPSK signal to the signal after processing with the exact hole-blowingmethod. The threshold for desired minimum power in this example wasselected to be 12 dB below RMS power. It can be seen that the exacthole-blowing method is highly effective in keeping the signal powerabove the desired minimum value. The I-Q plot of the QPSK signal afterhole-blowing is shown in FIG. 11. It can be seen that all traces havebeen pushed outside the desired limit. A “hole” appears in I-Q plotwhere none was evident before.

The instantaneous frequency of the modified signal is shown in FIG. 12.By comparison with FIG. 8, it can be seen that the instantaneousfrequency has been reduced from approximately 45× the symbol rate toapproximately 1.5× the symbol rate. Clearly the method has greatlyreduced the instantaneous frequency.

It should be noted that the exact hole-blowing method described here ishighly tolerant of timing error, but relatively intolerant of magnitudeand phase error (or equivalent errors in the correction factors c_(I)and c_(Q)). Timing error refers to the time at which the correctivepulse is added to the original signal. This tolerance to timing error isdue to the fact that the pulse shaping filter generally will have afairly broad magnitude peak relative to the duration of thelow-magnitude event. The effect of this sort of timing error is shown inFIG. 13, where a timing error of up to one-fourth of a symbol perioddegrades the AMR by only 1 dB.

Having shown that the exact hole-blowing method is effective in removinglow-magnitude events, the effect on in-band and out-of-band signalquality will now be described. The PSDs of the QPSK signal before andafter hole-blowing are indistinguishable in the frequency domain asshown in FIG. 14. The effect on EVM is illustrated in FIG. 15 a, showingan I-Q plot of the modified signal after matched-filtering andbaud-synchronous sampling. The result shows the message that a receiverwould obtain upon demodulation of the modified signal. It can be seenthat distortion has been introduced to the signal, in that not allsamples fall exactly on one of the four QPSK constellation points. TheRMS EVM can be defined as:

$\sigma_{evm} = \frac{\sqrt{\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{{a_{k} - {\hat{a}}_{k}}}^{2}}}}{\sqrt{\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{a_{k}}^{2}}}}$where a_(k) and â_(k) are, respectively, the ideal and actual PAMsymbols. For the particular example illustrated in FIG. 14, the RMS EVMwas calculated using N=16384 symbols and found to be 6.3%. Similarly,the peak EVM is defined as follows:

${p\; k_{evm}} = {\max\frac{{a_{k} - {\hat{a}}_{k}}}{\sqrt{\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{a_{k}}^{2}}}}}$

In this example, the peak EVM is found to be about 38%. If the signal isallowed to have greater magnitude dynamic range, the RMS EVM will belower. Recall, however, that increasing the allowed magnitude dynamicrange also increases the bandwidth and peak value of the instantaneousfrequency. Accordingly, a tradeoff can be made between the desiredsignal quality (EVM) and the instantaneous frequency requirements placedon a practical polar modulator.

To assure an imperceptible effect on the signal PSD (see FIG. 14), thechoice of the correction pulse shape should substantially match thesignal band limiting pulse shape. When the pulse oversampling rate isfour or more (samples per symbol time), the preceding algorithm can besimplified. Specifically, the calculation of t_(min) can be eliminated,and the correction pulse is inserted time-aligned with the existingsignal sample s(t1). As shown in FIG. 13, the maximum error in minimummagnitude is 1 dB for 4× oversampling, and decreases to 0.2 dB for 8×oversampling.

The difference in practice between an approach in which hole-blowing isperformed taking account of t_(min) and an approach in which thecalculation of t_(min) can be eliminated is illustrated in the examplesof FIGS. 15 b and 15 c, respectively. In the example of FIG. 15 b,hole-blowing was performed using a root-raised cosine pulse that was thesame as the pulse-shaping pulse. In the example of FIG. 15 c,hole-blowing was performed using a Hanning window for the correctionpulse, with a time duration equal to ½ the symbol duration. As may beseen, in FIG. 15 c, the original signal trajectory is changed as littleas possible is order to skirt the region of the hole. In many instances(if not most), however, the calculation of t_(min) can be eliminated (asin FIG. 15 b).

FIG. 16 a shows an arrangement for symbol-rate hole-blowing. A digitalmessage is applied to a main path in which pulse-shaping andupconversion are performed. An auxiliary path includes an analyzer blockthat produces a correction signal. An adder is provided in the main pathto sum together the main signal and the correction signal produced bythe analyzer.

Referring to FIG. 16 b, a particularly advantageous embodiment forperforming symbol-rate hole-blowing is shown. Two signal paths areprovided, a main path and an auxiliary path. Outputs of the main signalpath and the auxiliary signal path are summed to form the final outputsignal.

The main signal path receives symbols (or chips) and performspulse-shaping on those symbols (or chips), and is largely conventional.However, the main signal path includes a delay element used to achievesynchronization between the main signal path and the auxiliary signalpath.

In the auxiliary path, the symbols (or chips) are applied to acorrection DSP (which may be realized in hardware, firmware orsoftware). The correction DSP performs hole-blowing in accordance withthe exact method outlined above and as a result outputs an auxiliarystream of symbols (or chips). These symbols (or chips) will occur at thesame rate as the main stream of symbols (or chips) but will small inmagnitude in comparison, and will in fact be zero except when the signalof the main path enters or is near the hole. The relative timing of themain and auxiliary paths may be offset by T/2 such that the small-valuedsymbols of the auxiliary path occur at half-symbol timings of the mainsignal path.

In an exemplary embodiment, the correction DSP calculates the signalminimum with respect to every successive pair of symbols (or chips), bycalculating what the signal value would be corresponding to therespective symbols (or chips) and applying the locally linear model. Incalculating what the signal value would be corresponding to a symbol (orchip), the same pulse as in the main signal path and is applied to thatsymbol (or chip) and some number of previous and succeeding symbols (orchips). This use of the pulse to calculate what the value of the signalat a particular time will be following pulse-shaping is distinct fromthe usual pulse-shaping itself.

After the correction symbols (or chips) of the auxiliary signal pathhave been determined, they are pulse-shaped in like manner as those ofthe main signal path. The pulse-shaped output signals of the main andauxiliary paths are then combined to form the final output signal.

Since the duration of p(t) is finite (L), the signal can be evaluatedahead of time for the correlation between low-magnitude events and theinput symbol stream (L^(M) cases). The analyzer can then operate solelyon the {a_(i)} to determine the c_(i) and t_(i). (The correction symbolsc_(i) are very small, and have negligible effect on the message stream.)In this variant of symbol-rate hole-blowing, the calculated symbolsb_(m) are added to the original message stream, and the entire newstream is bandlimited once through the filter.

Iterative Methods

The foregoing description has focused on symbol-rate and sample-ratehole-blowing techniques. Although the term “exact” has been used torefer to these techniques, some inaccuracy and imprecision isinevitable. That is, the resulting signal may still impinge upon thedesired hole. Depending on the requirements of the particular system, itmay be necessary or desirable to further process the signal to removethese residual low-magnitude events. One approach it to simply specify abigger hole than is actually required, allowing for some margin oferror. Another approach is to perform hole-blowing repeatedly, oriteratively.

In the quadrature domain, sample-rate hole-blowing may be performed oneor multiple times (FIG. 16 c) and symbol-rate hole-blowing may beperformed one or multiple times (FIG. 16 d). In the latter case,however, at each iteration, the symbol rate is doubled. For example, ina first iteration, correction symbols may be inserted at T/2. In asecond iteration, correction symbols may be inserted at T/4 and 3T/4,etc. One or more iterations of symbol-rate hole-blowing may be followedby one or more iterations of sample-rate hole-blowing (FIG. 16 e). (Ingeneral, sample-rate hole-blowing cannot be followed by symbol-ratehole-blowing.)

Another alternative is sample-rate hole-blowing in the polar domain.

Polar-Domain Hole-Blowing

The foregoing quadrature-domain hole-blowing techniques preserve ACLRvery well at the expense of some degradation in EVM. Other techniquesmay exhibit a different tradeoff. For example, polar-domain hole-blowingpreserves EVM very well at the expense of ACLR. Therefore, in aparticular application, quadrature-domain hole-blowing, polar-domainhole-blowing, or a combination of both, may be applied.

The magnitude and phase components of a polar coordinate system can berelated to the in-phase and quadrature components of a rectangularcoordinate system as

${\rho(t)} = \sqrt{{s_{I}^{2}(t)} + {s_{Q}^{2}(t)}}$θ(t) = tan⁻¹(s_(Q)(t)/s_(I)(t))

FIG. 17 a is a block diagram of a portion of a radio transmitter inwhich polar-domain nonlinear filtering (i.e., “hole-blowing”) isperformed. The diagram shows how magnitude and phase as expressed in theforegoing equations are related to polar-domain nonlinear filtering andthe polar modulator.

In FIG. 17 a, G represents gain of the polar modulator. In operation,the digital message is first mapped into in-phase and quadraturecomponents in rectangular coordinate system. The in-phase and quadraturecomponents are converted into magnitude and phase-difference by arectangular-to-polar converter. By knowing the starting point of thephase and the phase-difference in time, the corresponding phase in timecan be calculated. This phase calculation is done at a later stage, inthe polar modulator. Before feeding the phase-difference to a polarmodulator, polar-domain nonlinear filtering is performed.

A more detailed block diagram is shown in FIG. 17 b, illustrating a mainsignal path and a correction signal path for two signal channels, ρ andθ. In the correction path, sequential values of ρ are used to perform asignal minimum calculation and comparison with a desired minimum. Ifcorrection is required based on the comparison results, then for eachchannel the magnitude of the required correction is calculated. A pulse(or pair of pulses in the case of the θ channel) is scaled according tothe required correction and added into the channel of the main path,which will have been delayed to allow time for the correction operationsto be performed.

An example of the magnitude and phase components (phase-difference) inpolar coordinates is shown in FIG. 18. As can be seen from the plot,when the magnitude dips, there is a corresponding spike (positive ornegative) in the phase-difference component.

The spike in the phase-difference component suggests that there is arapid phase change in the signal, since the phase and phase-differencehave the following relationship:θ(t)=∫{dot over (θ)}(t)dt

The dip in magnitude and the rapid phase change are highly undesirableas both peaks expand the bandwidth of each polar signal component. Thegoal of the polar domain nonlinear filtering is to reduce the dynamicrange of the amplitude swing as well as to reduce the instantaneousphase change. The dynamic range of the amplitude component and themaximum phase change a polar modulator can handle is limited by thehardware. Polar domain nonlinear filtering modifies the signal beforethe signal being processed by the polar modulator. This pre-processingis to ensure that the signal dynamic range will not exceed the limit ofthe implemented hardware capability of the polar modulator, andtherefore avoids unwanted signal distortion.

Nonlinear filtering in a polar coordinate system is somewhat morecomplex than nonlinear filtering in a rectangular coordinate system.Both components (magnitude and phase) in the polar coordinate systemhave to be handled with care to prevent severe signal degradation.

The following description sets forth how polar-domain nonlinearfiltering is performed in an exemplary embodiment. Polar-domainnonlinear filtering is composed of two parts. The first part isnonlinear filtering of magnitude component, and the second part isnonlinear filtering of the phase component (phase-difference). These twoparts will each be described in turn.

Nonlinear Filtering of Magnitude Component

If the magnitude dynamic range exceeds the capability of the polarmodulator, the output signal will be clipped. This clipping will resultin spectral re-growth and therefore greatly increase the adjacentchannel leakage ratio (ACLR). One method that can be applied to reducethe dynamic range of the magnitude component is hole-blowing (ornonlinear filtering).

The purpose of this nonlinear filtering of the magnitude component is toremove the low magnitude events from the input magnitude ρ(t), andtherefore reduce the dynamic range of the magnitude swing. Assume athreshold (TH_(mag)) for the minimum magnitude; by observing the signalρ(t), it is possible to obtain the time intervals where the signal fallsbelow the threshold. Assume there are N time intervals where the signalfalls below the threshold and that the magnitude minimums for each ofthese time intervals happen at t₁ . . . , t_(N), respectively.Therefore, the nonlinear filtering of magnitude component can beexpressed as:

${\hat{\rho}(t)} = {{\rho(t)} + {\sum\limits_{n = 1}^{N}{b_{n}{p_{n}\left( {t - t_{n}} \right)}}}}$where b_(n) and p_(n)(t) represent the inserted magnitude and pulse forthe n th time interval. The nonlinear filtered signal is composed of theoriginal signal and the inserted pulses. The magnitude of the insertedpulse is given as:b _(n) =TH _(mag)−ρ(t _(n))

The inserted pulse p_(n)(t) has to be carefully chosen so that thesignal degradation with respect to ACLR is minimized. It is desirablefor the pulse function to have smooth leading and trailing transitions.A suitable pulse (DZ3) is based on McCune's paper entitled “TheDerivative-Zeroed Pulse Function Family,” CIPIC Report #97-3, Universityof California, Davis, Calif., Jun. 29, 1997. The impulse response of theDZ3 pulse is plotted in FIG. 19.

Results of an example for nonlinear filtering of the magnitude componentare shown in FIG. 20. As can be seen from the plot, signal intervalswhere the magnitude dropped below the threshold are compensated by theinserted pulses. As a result, the dynamic range of the magnitude swingis reduced.

Nonlinear Filtering of Phase Component

In an exemplary embodiment, the input to the polar modulator isphase-difference, not phase. A voltage-controlled oscillator (VCO) inthe polar modulator integrates the phase-difference and produces thephase component at the output of the polar modulator. Thephase-difference is directly related to how fast the VCO has tointegrate. If the phase-difference exceeds the capability of the VCO,the output signal phase will lag (or lead) the actual signal phase. As aresult, phase jitter will occur if the VCO is constantly unable to keepup with the actual signal phase. This phase jitter will result inconstellation rotation, and may therefore severely degrade EVM.

The purpose of the nonlinear filtering of phase component is to suppresslarge (positive or negative) phase-difference events so that phaseaccumulation error will not occur. It is important to know that theoutput of the VCO is an accumulation of the input (phase-difference).Therefore, any additional processing to the phase-difference has toensure that the phase error will not accumulate. The nonlinear filteringof the phase component described presently carefully modifies thephase-difference in a way that the accumulated phase might deviate fromthe original phase trajectory from time to time. However, after acertain time interval, it will always merge back to the original phasepath.

Nonlinear filtering of the phase component is done by first finding thelocation where the absolute value of the phase-difference is beyond theintegral capability of the VCO. Assume there are a total of M eventswhere the absolute values of the phase-difference are beyond thecapability of the VCO, and that the peak absolute value for each eventhappens at t_(m). Nonlinear filtering of the phase-difference componentmay then be described by the following expression:

${\overset{\overset{\Cap}{.}}{\theta}(t)} = {{\overset{\overset{\Cap}{.}}{\theta}(t)} + {\sum\limits_{m = 1}^{M}{c_{m}{p_{p,m}\left( {t - t_{m}} \right)}}}}$where p_(p,m)(t) is the pulse inserted to the phase-difference componentat time t_(m), and c _(m) is the corresponding magnitude given byc _(m) =TH _(p)−{dot over (θ)}(t _(m))where TH_(p) is the threshold for the phase-difference.The inserted pulse should satisfy the following equation∫p _(p,m)(t)dt=0

The result of inserting a pulse is basically the same as altering thephase trajectory. However, by inserting pulses that satisfy the aboveequation, the modified phase trajectory will eventually merge back tothe original phase trajectory. This can be seen by the followingequation:

${\int{{\overset{\overset{\Cap}{.}}{\theta}(t)}{\mathbb{d}t}}} = {{\int{{\overset{.}{\theta}(t)}{\mathbb{d}t}}} + {\sum\limits_{m = 1}^{M}{c_{m}{\int{{p_{p,m}(t)}{\mathbb{d}t}}}}}}$The second term on the right-hand side will eventually disappear.Therefore, the modified phase trajectory will merge back to the originalphase trajectory.

An example of an added pulse p_(p)(t) suitable for nonlinear filteringof the phase component is plotted in FIG. 21. Again, it is veryimportant for the pulse function to have smooth leading and trailingtransitions. The pulse used for nonlinear filtering at thephase-difference path is composed of two pulses. These two pulses havethe same area but different polarity and durations. Therefore, theintegration of the combined pulse with respect to time is zero.

Results of an example of nonlinear filtering of the phase-differencecomponent are shown in FIG. 22. As can be seen from the plot, themodified phase trajectory has a smoother trajectory than the originalphase trajectory. The modified phase trajectory requires less bandwidthfrom the VCO than the original phase trajectory.

Polar-domain nonlinear filtering is composed of the filtering ofmagnitude and phase components. If the nonlinear filtering is donejointly (both magnitude and phase), better spectral roll-off can beachieved. However, each nonlinear filtering operation can also be doneindependently.

Instead of nonlinear filtering of the phase-difference component, directnonlinear filtering of the phase component can also be implemented. FIG.23 shows an example in which the original signal has a steep increase inphase from time t₁ to t₂. One way of reducing the phase change is byinterpolating between the straight line v(t) and the original phasepath. The interpolation can be expressed as:{circumflex over (θ)}(t)=w(t)θ(t)+(1−w(t))v(t)where w(t) is a weighting factor. The weighting factor can be a constantor may vary in accordance with DZ3, a Gaussian function, or the like.Again, it is important for the weighting function to have smooth leadingand trailing edges.

Polar-domain nonlinear filtering can be used in concatenation withquadrature-domain nonlinear filtering. Such an approach may involve lesscomputational complexity than iterative quadrature-domain nonlinearfiltering.

If only one iteration of quadrature-domain nonlinear filtering isperformed, low-magnitude, high-phase-variation events may still occur,but with low probability. These low probability events can degradesignal quality if not properly managed. To eliminate theselow-probability events, the foregoing polar-domain nonlinear filteringmethod may be used following a single iteration of quadrature-domainnonlinear filtering, maintaining low EVM and low ACLR. FIG. 24 shows theblock diagram for this concatenated system. FIG. 25 shows the PSD of theoutput signal processed by the quadrature-polar nonlinear filteringmodule. Curve A represents the signal with eight iterations ofquadrature-domain non-linear filtering. Curve B represents the signalwith one iteration of quadrature-domain nonlinear filtering followed byone iteration of polar-domain nonlinear filtering. Spectral re-growth isbelow −60 dB. In addition, good spectral roll-off is achieved.

Reduction of Computational Load for Real-Time Applications

In a practical polar modulator, the hole-blowing algorithm must beimplemented in real-time using digital hardware and/or software.Real-time implementation of the exact hole-blowing method presentsseveral challenges. The particular challenges faced are, to some extent,dependent on the overall architecture selected. There are (at least) twoalternative architectures for implementing the exact hole-blowingalgorithm. In the first architecture, referred to as symbol-ratehole-blowing, corrective impulses are calculated and added to the datastream with appropriate timing, after which pulse shaping is performed.In the second architecture, referred to as sample-rate (or oversampled)hole-blowing, the full pulse-shaped signal is calculated, after whichweighted pulses are added to the pulse-shaped signal.

Because of its computational complexity, the exact hole-blowingalgorithm described earlier is difficult to implement using digitalhardware. In the exact hole-blowing algorithm, arithmetic division,square, and square-root operations are required. These arithmeticoperations greatly increase the complexity of the digital hardware andshould be avoided if possible. Reduction in computational complexity ofthe implemented algorithm leads directly to a reduction in hardwarecomplexity. The real-time hole-blowing algorithm described hereinrequires no division, square, or square-root operations.

Although a real-time hole-blowing algorithm can be implemented in eithersymbol-rate or sample-rate form, it is generally more desirable toimplement symbol-rate hole-blowing. In particular, symbol-ratehole-blowing algorithm allows the digital hardware to be operated at aslower clock speed and does not require as much memory to perform pulseinsertion. A symbol-rate real-time hole-blowing algorithm will thereforebe described.

Generally speaking, real time hole-blowing algorithm is very similar tothe exact hole-blowing algorithm. However, some assumptions andapproximations are made in real time hole-blowing algorithm to simplifyimplementation. In the real-time hole-blowing algorithm, assumptions aremade regarding where magnitude minimums are most likely to occur basedon the structure of the signal constellation(s). The assumption allowsfor determination of where to insert the pulse, if needed, withoutcalculating the entire signal magnitude. In addition, the arithmeticoperations in exact bole-blowing algorithm are greatly simplified bynormalizing the expression √{square root over (Δx²+Δy²)} to one. Thisnormalization eliminates the need for division, square, and square-rootoperations.

The differences between the real-time and exact hole-blowing algorithmsmay be appreciated by considering answers to the following questions:

1. What are the potential timings for the low-magnitude events?

2. What are the values of magnitude minimums, and how can they becalculated efficiently and accurately?

3. What are the in-phase and quadrature correction weights of themagnitude minimum if a signal falls below the prescribed threshold?

These questions will be addressed using the UMTS signal as an example.

1. What are the Potential Timings for the Low-Magnitude Events?

If the timings of the magnitude minimums can be estimated based on theinput data bits, then it is not necessary to calculate the wholewaveform in order to obtain the magnitude minimums. This shortcutaffords a great saving in computation. The locally linear model can beused to calculate the minimums with great accuracy if the approximatetiming for low-magnitude events is known.

As previously described, the minimum magnitudes for a bandlimited QPSKsignal usually occur close to half-symbol instance (nT+T/2). Thisassumption is supported by the histogram in FIG. 6. This assumption canalso be applied to higher-order pulse-shaped PSK signals whoseconstellation points lie on the same circle. A good example for thistype of signal is the UMTS signal with one active data channel. FIG. 26shows an example of the UMTS signal constellations with one active datachannel. As can be seen from the plot, all constellation points lie onthe same circle. If a histogram is constructed for this particularsignal, it may be seen that the timings for the low-magnitude events arevery likely to happen at every half-symbol instance. Therefore, for thisparticular signal, it is assumed that the timings for the low-magnitudeevents will occur at time nT+T/2, where n is an integer.

The above assumption will not be true in the case of a more complexsignal constellation. FIG. 27 shows an example of the UMTS signalconstellations with two active data channels. As can be seen from theplot, not all the constellation points lie on the same circle. Thealgorithm used to find the magnitude minimums for a more complex signalconstellation is illustrated in FIG. 28 a, and described as follows:

1. If the signal transitions from constellation point P1 at time nT andends at point P2 at time (n+1)T, a straight line is drawn connecting thetwo points. This line is denoted as the first line.

2. A second line is drawn perpendicular to the line that connects P1 andP2. The second line includes the origin and intersects the first line atpoint M.

3. The second line divides the first line into two sections whose lengthare proportional to D1 and D2, respectively. The timing for thelow-magnitude event is approximately nT+D1/(D1+D2)T.

4. If the second line and the first line do not intersect, there is noneed for pulse insertion.

5. Having determined where the magnitude minimums are most likely tooccur, the locally linear model is then used to calculate the localmagnitude minimums.

With the above algorithm, a look-up table can be built for differentsignal constellations. The size of the look-up table can be reduced ifthe constellation points are symmetrical to both x and y axis. A specialcase of the foregoing algorithm occurs where D1 always equal to D2,corresponding to the UMTS signal with one active data channel.Therefore, the magnitude minimums for one active data channel alwayshappen close to time nT+T/2. The above algorithm for finding theapproximate locations of magnitude minimums can also be generalized tosignals with more complex constellations.

FIG. 28 b shows the probability density function, by sample interval(assuming 15 samples per symbol), of minimum magnitude events wherecorrection is required, for both the exact method described earlier andthe real-time approximation just described. Note the closecorrespondence between the two functions.

2. What are the Values of the Magnitude Minimums, and how can they beCalculated Efficiently and Accurately?

With the knowledge of the timing where the magnitude minimums are likelyto occur, the locally linear model is then used to calculate themagnitude minimums. A determination must be made whether the magnitudeof the signal falls below the prescribed threshold. If the magnitudefalls below the prescribed threshold, in-phase and quadrature correctionweights, c_(I) and c_(Q), for the inserted pulse have to be calculated.

From the exact hole-blowing algorithm, the magnitude minimum can becalculated by the following equation:

$\rho_{m\; i\; n} = \frac{{{y\; 1\;\Delta\; x} - {x\; 1\;\Delta\; y}}}{\sqrt{{\Delta\; x^{2}} + {\Delta\; y^{2}}}}$where x1 and y1 are the in-phase and quadrature samples of the signal attime t1, and t1 is the time where the magnitude minimums are most likelyto occur.

The equation above involves one division, one square-root operation, twosquare operations, as well as several other operations—multiplication,addition and subtraction. The computational complexity can be reduced bynormalizing the denominator of the above equation.

Let the vector (Δx, Δy) be expressed as:Δx=ρxy cos(θ_(xy))Δy=ρ _(xy) sin(θ_(xy))where ρ_(xy) and θ_(xy) are the magnitude and phase of the vector (Δx,Δy). Then, the minimum magnitude of the signal can be expressed as:

$\rho_{m\; i\; n} = {\frac{{{y\; 1\;\Delta\; x} - {x\; 1\Delta\; y}}}{\sqrt{{\Delta\; x^{2}} + {\Delta\; y^{2}}}}\mspace{45mu} = {\frac{\rho_{xy}{{{y\; 1\;\cos\;\left( \theta_{xy} \right)} - {x\; 1\;{\sin\left( \theta_{xy} \right)}}}}}{\rho_{xy}\sqrt{{\cos^{2}\left( \theta_{xy} \right)} + {\sin^{2}\left( \theta_{xy} \right)}}}\mspace{45mu} = {{{y\; 1\;\cos\;\left( \theta_{xy} \right)} - {x\; 1\;\sin\;\left( \theta_{xy} \right)}}}}}$

With this normalization, no division, square, or square-root operationis required. In addition, it is not necessary to know ρ_(xy). However,the values of sin(θ_(xy)) and cos(θ_(xy)) are needed. Therefore, given avector (Δx, Δy), a way is needed to obtain (sin(θ_(xy)), cos(θ_(xy))efficiently.

Two ways will be described to evaluate sin(θ_(xy)) and cos(θ_(xy)) withrelatively low hardware complexity. The first one uses a line-comparisonmethod, and the second one uses the CORDIC (Coordinate Rotation forDigital Computer) algorithm.

First, consider the approximation of sin(θ_(xy)) and cos(θ_(xy)) using aline-comparison method. As can be seen from FIG. 29, lines with afunction of y=Mx partition the first quadrant into several sub-sections.By comparing the point (|Δx|,|Δy|) against the lines y=Mx, thesub-section that the point (|Δx|,|Δy|) falls into can be determined. Anypoint within a certain sub-section is represented by a pre-normalizedpoint (sin(θ_(xy) ^(i)), cos(θ_(xy) ^(i))), where i denotes the sectionthe point (|Δx|,|Δy|) belongs to.

The details of the algorithm are described as follows:

1. First, convert the vector (|Δx|,|Δy|) into the first quadrant(|Δx|,|Δy|).

2. Compare |Δy| with M|Δx| where M is positive.

3. If |Δy| is greater than M|Δx|, then (|Δx|,|Δy|) is located at theleft side of the line y=Mx

4. If |Δy| is smaller than M|Δx|, then (|Δx|,|Δy|) is located at theright side of the line y=Mx.

5. Based on the above comparisons with different lines, the sub-sectionthe point (|Δx|,|Δy|) belongs to can be located. Assuming the pointbelongs to section i, then the vector (sin(θ_(xy)), cos(θxy)) can beapproximated by the vector (sign(Δx)*sin(θ_(xy) ^(i)),sign(Δy)*cos(θ_(xy) ^(i))).

A table is used to store the pre-calculated values of(sin(θ_(xy) ^(i)),cos(θ_(xy) ^(i))). If a total of W lines are used for comparison, therewill be (W+1) sub-sections in the first quadrant. As a result, the wholevector plane is being divided into 4*(W+1) sub-sections.

The second method used to approximate (sin(θ_(xy)),cos(θ_(xy))) is aCORDIC-like algorithm. This method is similar to the line-comparisonmethod. However, the CORDIC-like algorithm partitions the vector planemore equally. A more detailed description of the algorithm is presentedas follows and is illustrated in FIG. 30:

1. First, convert the vector (Δx, Δy) into the first quadrant(|Δx|,|Δy|).

2. Second, rotate the vector (|Δx|,|Δy|) clock-wise with an angleΘ₀=tan⁻¹(i). The vector after the angle rotation is denoted as(|Δx|₀,|Δy|₀).

3. Let i=1.

4. If |Δy|_(i−1) is greater than 0, rotate the vector (|Δx|₀,|Δy|₀)clockwise with an angle Θ₀=tan⁻¹(2^(−i)). Otherwise, rotate the vectorcounter-clockwise with an angle of Θ_(i). The vector after the anglerotation is (|Δx|_(i),|Δy|_(i))

5. Let i=i+1.

6. Repeat steps 4 and 5 if needed.

7. Assume a total of K vector rotations are performed. This algorithmpartitions the first quadrant into 2^k sub-sections. The sign of |Δy|₀,|Δy|₁, . . . , |Δy|_(K−1) can be used to determine which of the 2^ksub-sections the vector (|Δx|,|Δy|) belongs to.

8. A table filled with pre-quantized and normalized values is then usedto approximate the vector (|Δx|,|Δy|).

9. If the look-up table gives a vector of (sin(θ_(xy) ^(i)), cos(θ_(xy)^(i))) for vector (|Δx|,|Δy|), then (sin(θ_(xy)), cos(θ_(xy))) can beapproximated by the vector (sign(Δx)*sin(θ_(xy) ^(i)),sign(Δy)*cos(θ_(xy) ^(i))).

The vector rotation in the CORDIC algorithm is carefully done so thatthe vector rotation is achieved by arithmetic shifts only. This leads toa very efficient structure. The accuracy of the approximation can beimproved by going through more CORDIC iterations. If a total of twovector rotations are performed, the resulting partition of the firstquadrant is similar as in the line-comparison method shown in FIG. 29.

The normalization of the expression √{square root over (Δx²+Δy²)} andthe efficient algorithms to approximate (sin(θ_(xy)), cos(θ_(xy)))greatly reduce the computational complexity of the locally minimummethod. Using these methods, ρ_(min) may be readily evaluated. If themagnitude minimum ρ_(min) falls below the threshold ρ_(desired), a pulseinsertion will be performed based on the hole-blowing algorithm. Thisleads to the third question:

3. What are the in-Phase and Quadrature Correction Weights if theMagnitude Minimum of the Signal Falls Below the Prescribed Threshold?

Similar techniques can be applied to the calculation of sin(θ) andcos(θ) by replacing Δx with Δx=ρ_(xy) cos(θ_(xy)) and Δy with Δy=ρ_(xy)sin(θ_(xy)), as follows:

$\begin{matrix}{{\sin\;\theta} = \frac{\Delta\; x\mspace{11mu}{sign}\mspace{11mu}\left( {{y\; 1\Delta\; x} - {x\; 1\Delta\; y}} \right)}{\sqrt{{\Delta\; x^{2}} + {\Delta\; y^{2}}}}} \\{= \frac{\rho_{xy}{\cos\left( \theta_{xy} \right)}\mspace{11mu}{sign}\mspace{11mu}\left( {{y\; 1\;\Delta\; x} - {x\; 1\;\Delta\; y}} \right)}{\rho_{xy}\sqrt{{{\cos\;}^{2}\left( \theta_{xy} \right)} + {\sin^{2}\left( \theta_{xy} \right)}}}} \\{= {\cos\left( {\theta_{xy}\;{{sign}\left( {{y\; 1\;\Delta\; x} - {x\; 1\;\Delta\; y}} \right)}} \right)}} \\{{\cos\;\theta} = {{- {\sin\left( \theta_{xy} \right)}}\;{sign}\;\left( {{y\; 1\;\Delta\; x} - {x\; 1\Delta\; y}} \right)}}\end{matrix}$

With the above simplification, the in-phase and quadrature correctionweights, c_(I) and c_(Q) can be calculated easily with simplesubtraction and multiplication:c _(I)=(ρ_(desired)−ρ_(min))cos θc _(Q)=(ρ_(desired)−ρ_(min))sin θSummary of the Real Time Hole-Blowing Algorithm

To facilitate real-time implementation, a novel approach is taken toestimate the timings for the low-magnitude events. Also, thelocally-minimum method is greatly simplified by normalizing theexpression √{square root over (Δx²+Δy²)}. As a result, no division,square operation, nor square-root operation are required.

Two methods are being proposed to evaluate (sin(θ_(xy)), cos(θ_(xy)))given the vector (Δx, Δy). The first one is the line-comparison method,and the second one is the CORDIC-like algorithm. The implementationcomplexities of the above two methods are generally low since onlyarithmetic shifts and comparisons are needed.

In the exemplary embodiments described above, it is assumed thatlow-magnitude events in the complex baseband signal s(t) correspond to,i.e., correlate in time with, high-frequency events. While such anassumption is acceptable in some applications, in others that employmodulation formats resulting in signals that spend a significant amountof time near the origin in the I-Q signal plane (like those that map tosignal constellations having constellation points near the origin), theassumption is not acceptable. In fact, in some applications somelow-magnitude events do not involve high-frequency content, even thoughthe low-magnitude events have magnitudes falling below the low-magnitudethreshold α. This is illustrated in FIG. 37, which is a waveform diagramof the magnitude component ρ of a polar-coordinate baseband signal.While the magnitude of the magnitude component ρ changes abrupt duringevents 1 and 2, indicative of high-frequency events, the magnitudechanges slowly during event 3, even though the magnitude remains belowthe low-magnitude threshold α. Hence, performing hole blowing orpolar-domain nonlinear filtering based on detection of low-magnitudeevents can lead to undesirable results.

In some applications, using the magnitude component p to inferoccurrences of high-frequency events can also result in thenon-detection of high-frequency events. In particular, depending on themodulation format being employed, high-frequency events can occur in themagnitude component ρ even during times when the magnitude of themagnitude component ρ remains above the low-magnitude threshold α. Insuch circumstances, the higher magnitude high-frequency events goundetected and correction pulses are not generated and inserted whenthey are actually needed.

To overcome the limitations of using the magnitude component ρ to inferoccurrences of high-frequency events, according to one embodiment of theinvention high-frequency events are detected by monitoring thephase-difference component Δθ, instead, as illustrated in FIGS. 38 and39. According to this embodiment of the invention, a CORDIC converter3900 first converts the I and Q components representing the complexbaseband signal s(t) into polar coordinates ρ and θ. After the phasecomponent θ is converted into a phase-difference component Δθ by aphase-difference calculator 3902, a threshold comparator 3904 identifiestime instances or time intervals (i.e., events) in which thephase-difference component Δθ rises above or falls below upper and lowerphase-difference thresholds β and −β. The upper and lowerphase-difference thresholds β and −β are determined and set beforehandbased on an in-band performance criterion, an out-of-band performancecriterion, or a balance or combination of in-band and out-of-bandperformance criteria, such as a balance or combination of acceptableACLR and maximum allowable EVM.

Similar to as in the exemplary embodiments described above, thedetection process of the present embodiment is performed by a DSPimplemented in hardware, firmware, software or any combination ofhardware, firmware and software. If a high-frequency event is detectedin the phase-difference component Δθ, one or more correction pulses aregenerated and inserted in the I and Q components of the complex basebandsignal s(t) in the temporal vicinity of the detected high-frequencyevent (e.g., using one of the hole blowing techniques described above),or are generated and inserted in the magnitude component p and/orphase-difference component Δθ in the temporal vicinity of the detectedhigh-frequency event in the polar domain (e.g., similar to thepolar-domain nonlinear filtering techniques described above). In otherwords, detection of high-frequency events in the phase-differencecomponent Δθ is used as a basis for performing hole blowing in thequadrature domain, performing nonlinear filtering of either or both themagnitude component p and phase-difference component Δθ in the polardomain, or applying quadrature-domain hole blowing in concatenation withpolar-domain nonlinear filtering. If hole blowing is performed, it maybe performed at either the sample rate or the symbol rate and eitherprior to or after pulse-shape filtering.

FIGS. 40 and 41 illustrate an alternative high-frequency detectionapproach. According to this embodiment of the invention, first andsecond threshold comparators 4102 and 4104 are employed to detectoccurrences of high-frequency events by monitoring time instances orintervals in which both the magnitude of the magnitude component ρ fallsbelow a predetermined low-magnitude threshold α and the phase-differencecomponent Δθ rises above or falls below upper and lower phase-differencethresholds β and −β. For example, events 1 and 2 in FIG. 40 areconsidered to include high-frequency content since both the magnitude ofthe magnitude component ρ falls below the low-magnitude threshold α andthe phase-difference of the phase-difference component Δθ rises abovethe upper phase-difference thresholds β (event 1) or falls below thelower phase-difference threshold −β (event 2). Similar to as in thepreviously described embodiment, the detection process of the presentembodiment is performed by a DSP implemented in hardware, firmware,software or any combination of hardware, firmware and software. Further,and similar to the previously described embodiment, the thresholds α, βand −β are determined and set beforehand based on an in-band performancecriterion, an out-of-band performance criterion, or a balance orcombination of in-band and out-of-band performance requirements, such asa balance or combination of acceptable ACLR and maximum allowable EVM.

Based on detection of a high-frequency event in the magnitude andphase-difference components ρ and Δθ, one or more correction pulses aregenerated and inserted in the I and Q components of the complex basebandsignal s(t) in the quadrature domain in the temporal vicinity of thedetected high-frequency event (e.g., using one of the hole blowingtechniques described above), or are generated and inserted in themagnitude component ρ and/or phase-difference component Δθ in thetemporal vicinity of the detected high-frequency event in the polardomain (e.g., similar to the polar-domain nonlinear filtering techniquesdescribed above). In other words, detection of high-frequency events inthe magnitude and phase-difference components ρ and Δθ is used as abasis for performing hole blowing in the quadrature domain, performingnonlinear filtering of either or both the magnitude component ρ andphase-difference component Δθ in the polar domain, or applyingquadrature-domain hole blowing in concatenation with polar-domainnonlinear filtering. If hole blowing is performed, it may be performedat either the sample rate or the symbol rate and either prior to orafter pulse-shape filtering.

APPENDIX 1 A1.0 Analysis of the O'Dea Hole-Blowing Methods

In this Appendix, a detailed analysis is presented of the hole-blowingmethods proposed in U.S. Pat. Nos. 5,696,794 and 5,805,640. These twopatents are very similar and are therefore treated simultaneously. Themain difference is that the former patent (U.S. Pat. No. 5,696,794)modifies the symbols to be transmitted, while the latter patent (U.S.Pat. No. 5,805,640) adds pulses at T/2 symbol timing instants. Forbrevity, U.S. Pat. No. 5,696,794 will be referred to as the symbol ratemethod, and U.S. Pat. No. 5,805,640 will be referred to as the T/2method. An overview of both methods is first presented, followed by anexamination of performance with two different signal modulations. Thefirst test signal is or π/4 QPSK with zero-ISI raised cosine pulseshaping. This is the modulation employed in the two patents. The secondtest signal is a UMTS 3GPP uplink signal with one active DPDCH and aDPDCH/DPCCH amplitude ratio of 7/15. UMTS uses square-root raised-cosinepulse shaping with 0.22 rolloff characteristic.

A1.1 Overview of the O'Dea Hole-Blowing Algorithms

The term “half-symbol timing” is defined to be those time instants thatare exactly halfway between symbol times. That is, if the PAM signal ismodeled as

${s(t)} = {\sum\limits_{k}{a_{k}{p\left( {t - {kT}} \right)}}}$where T is the symbol period and p(t) is the pulse shape, then thehalf-symbol times correspond to t=kT+T/2 where k is an integer. Forclarity of presentation, it will be assumed that the maximum value ofp(t) has been normalized to unity.

Both methods test for the existence of undesirable low-power events bymeasuring the signal magnitude at the half-symbol time instants, andcomparing this value to some desired minimum magnitude mag_d:mag _(—) s=|s(kT+T/2)|≦vmag _(—) d

Both patents use the same method to calculate the phase of thecorrective pulse(s). Assume that the low-magnitude event occurs betweensymbols k and k+1. First, determine the so-called phase rotationθ_(rot), which is simply the change in phase in the transition fromsymbol k to symbol k+1, as illustrated in FIG. 31. The corrective phaseθ_(adj) is given by

$\theta_{adj} = {\theta_{k} + \frac{\theta_{rot}}{2}}$Where θ_(k) is the phase of the kth symbol. A vector with phase equal tothe adjustment phase is orthogonal to a straight line drawn from symbolk to symbol k+1, as illustrated in FIG. 31. Note that since there are afinite number of possible phase rotations, there are also a finitenumber of possible phase adjustments, so that explicit calculation ofthe rotation phase is not necessary.

If the symbol rate approach (U.S. Pat. No. 5,696,794) is used, the twosymbols adjoining the low-magnitude event (i.e., symbols k and k+1) aremodified by the addition of a complex scalar. The magnitude of thiscomplex scalar is given bym=0.5(mag _(—) d−mag _(—) s)/p _(mid)where p_(mid) is the amplitude of the pulse shaping filter at t=T/2.(The rationale for calculating the magnitude of the correction in thismanner, however, is unclear.) The complex scalar adjustment is thenc _(adj) =m exp(jθ _(adj))and the resulting modified symbols are given by{tilde over (a)}_(k) =a _(k) +c _(adj){tilde over (a)}_(k+1) =a _(c+1) +c _(adj)

Note that both symbols are modified in the same manner.

Thus, a noise component is added to the signal by intentional relocationof the information symbols {a_(k)}. This can “confuse” any equalizer atthe receiver, which will expect that any signal distortion is due to thechannel.

If the T/2 approach (U.S. Pat. No. 5,805,640) is used, a complex scalaris added to the symbol stream, before pulse-shaping, at the appropriatehalf-symbol time instants. When a low magnitude event is detected at tkT+T/2, a complex symbol with magnitudem=(mag _(—) d−mag _(—) s)and phase equal to θ_(adj) (given above) is added at t=kT+T/2. Thisrestriction to T/2 insertion timing limits this method to circularsignal constellations.

One additional difference between the symbol rate method and the T/2method is that the symbol rate method is intended to be appliediteratively until the signal magnitude does not drop below somethreshold. There is no mention of an iterative process in the T/2patent.

A1.2 Performance with π/4 OPSK

Consider now the performance of the known hole-blowing algorithmsrelative to the disclosed “exact” hole-blowing method when the targetedsignal is π/4 QPSK. It is interesting to note that this signal has a“hole” in its constellation if the rolloff is high, e.g., α=0.5. Arolloff of 0.22 was chosen so that the signal would not have apre-existing hole.

FIG. 32 shows the CDF obtained from the disclosed method and the twoknown methods with the aforementioned p/4 QPSK signal. The desiredminimum power level was set at 9 dB below RMS. These simulation resultsare based on 16384 symbols with 32 samples/symbol. The figure clearlyshows that the exact method is much more effective than either of theknown methods. It is also evident that both known methods performsimilarly. This is not surprising given the similarity of the twoapproaches.

Some explanation of the performance of the known methods as shown inFIG. 32 is in order. It was noted earlier that there are sources oferror possible in calculation of both the magnitude and phase of thecorrection pulses. FIG. 33 shows an example where the prior-art symbolrate method works fairly well. The signal envelope is not pushedcompletely out of the desired hole, but the method is performingmore-or-less as intended. In contrast, FIG. 34 shows an example wherethe prior-art symbol rate method does not perform well. This exampleillustrates error in both the magnitude and phase of the correctivepulses. In this example, the trace passes on the “wrong side” of theorigin relative to the assumptions made in calculating the correctionphase. Thus the trace is pushed in the wrong direction. Furthermore, itis evident from this example that the magnitude at T/2 is not theminimum magnitude, so that even if the phase had been calculatedcorrectly, the signal would not have been pushed far enough.

FIG. 35 shows an example where the prior-art T/2 method does not performwell. (The segment shown in FIG. 35 is the same segment of the signalshown in FIG. 34, where the symbol rate method did not perform well.)Comparing FIG. 34 and FIG. 35, it is clear that the two methods yieldnearly identical traces. It can be seen that the symbol rate method onlyalters the symbols adjacent to the low-magnitude event, while the T/2method affects more symbols.

A1.3 Performance with a 3GPP Uplink Signal

Consider now the performance of the known methods with a more realisticsignal, that being a 3GPP uplink signal with one active DPDCH and anamplitude ratio of 7/15. FIG. 36 shows the CDF's obtained with thedisclosed exact method and with the known hole-blowing methods whenapplied to one frame (38400 chips) of the signal with 32 samples/chip.It is clear that the exact correction method greatly outperforms theknown methods, and that the known methods perform comparably.

1. A method of processing a communications signal, comprising: setting ahigh-frequency threshold with a frequency detecting unit; convertingin-phase and quadrature components of a baseband signal to magnitude andphase-difference components with a converter unit; detectinghigh-frequency events in said phase-difference component during which arate of change of phase exceeds said high-frequency threshold with thefrequency detecting unit; altering a signal trajectory of said basebandsignal, in response to detected high-frequency events; upconverting saidbaseband signal after the signal trajectory of said baseband signal hasbeen altered; coupling an upconverted phase-difference component to afirst input of a modulator; and coupling said magnitude component to asecond input of said modulator, wherein altering the signal trajectoryof said baseband signal comprises nonlinear filtering of said magnitudecomponent and nonlinear filtering of said phase-difference component. 2.The method of claim 1 wherein said high-frequency threshold has afrequency based on one of an in-band performance criterion and anout-of-band performance criterion.
 3. The method of claim 1 whereinaltering the signal trajectory of said baseband signal comprisesperforming a hole blowing process on said in-phase and quadraturecomponents.
 4. The method of claim 1 wherein altering the signaltrajectory of said baseband signal comprises nonlinear filtering of saidphase-difference component.
 5. The method of claim 4 wherein nonlinearfiltering said phase-difference component comprises reducing a rate ofchange of phase of said baseband signal along the signal trajectory ofsaid baseband signal.
 6. The method of claim 1 wherein altering thesignal trajectory of said baseband signal comprises adjusting themagnitude of said baseband signal along the signal trajectory of saidbaseband signal.
 7. The method of claim 1 wherein said high-frequencythreshold has a frequency based on a balance or combination of in-bandand out-of-band performance criteria.
 8. The method of claim 1, furthercomprising: setting a low-magnitude threshold; detecting low-magnitudeevents in said magnitude component during which the magnitude of saidmagnitude component falls below said low-magnitude threshold; andaltering the signal trajectory of said baseband signal, in response todetected low-magnitude events that correlate in time with respect todetected high-frequency events.
 9. An apparatus for conditioning acommunications signal, comprising: a rectangular-to-polar converterconfigured to convert in-phase and quadrature components of a basebandsignal to magnitude and phase components; a phase-difference calculatorconfigured to convert said phase component to a phase-differencecomponent; frequency detecting means configured to detect high-frequencyevents in said phase-difference component having a rate of change ofphase exceeding a high-frequency threshold; means for altering a signaltrajectory of said baseband signal, in response to detectedhigh-frequency events; and a modulator having a first input configuredto receive said magnitude component and a second input configured toreceive an upconverted phase-difference component, after said means foraltering the signal trajectory has altered the signal trajectory of saidbaseband signal.
 10. The apparatus of claim 9 wherein saidhigh-frequency threshold has a frequency based on either an in-bandperformance criterion or an out-of-band performance criterion.
 11. Theapparatus of claim 10 wherein said means for altering the signaltrajectory of said baseband signal comprises means for alteringmagnitudes of said in-phase and quadrature components.
 12. The apparatusof claim 9 wherein said means for altering the signal trajectory of saidbaseband signal comprises a nonlinear filter configured to alter a rateof change of phase of said phase-difference component.
 13. The apparatusof claim 9 wherein said means for altering the signal trajectory of saidbaseband signal comprises a nonlinear filter configured to alter themagnitude of said magnitude component.
 14. The apparatus of claim 13wherein said means for altering the signal trajectory of said basebandsignal further comprises a nonlinear filter configured to alter thefrequency of said phase-difference component.
 15. The apparatus of claim9, further comprising magnitude detecting means configured to detectlow-magnitude events in said magnitude component having magnitudesfalling below a low-magnitude threshold.
 16. The apparatus of claim 15wherein said means for altering the signal trajectory of said basebandsignal is configured to alter the signal trajectory of said basebandsignal, in response to detected low-magnitude events in said magnitudecomponent that correlate in time with detected high-frequency events insaid phase-difference component.
 17. The apparatus of claim 9 whereinsaid high-frequency threshold has a frequency based on a balance orcombination of in-band and out-of-band performance criteria, instead ofon said in-band performance criterion or said out-of-band performancecriterion.